Pseudo-Euclidean Hurwitz pairs and the Kaluza-Klein theories

Ławrynowicz, Julian; Rembieliński, Jakub

doi:10.1088/0305-4470/20/17/018

Cited by 27 publications

(10 citation statements)

References 11 publications

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“…where a (1), o~ (5) ate the idenlification mappings in (11) and (12). P r o o f. We compare the equations (10) and in Proposition 6, and see the difference at a3 and icr3.…”

Section: (G74mentioning

confidence: 96%

Hurwitz duality theorems for Fueter and Firac equations

Ławrynowicz

Сузукі

1997

AACA

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Abstract. The paper aims at proving general duality theorems for Hermitian Hurwitz pairs (abbreviated as HHP), related to the generalized Dirac or Fueter equations, and gives comparison theorems for solutions of #-self adjoint Dirac or Fueter equations, where #-self adjointness implies the self axljointness with respect to certain indefinite metrics. In low-dimensional cases these duality theorems can be described in a more explicit m 91 Namely, we can get dualities between the Minkowski spacetime with signature (+,---) and the neutral space with signature (++,--) related to the Penrose theory. We can also get the dualities between the Neveu-Schwarz model with (+,-) and the holomorphic-antiholomorphic model with (-,-). These dualities can be described in terms of the Wick rotations. We notice that the Wick rotation cannot be used directly because our Dirac operators ate #-self adjoint, while the Wick rotations break down this property.

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“…where a (1), o~ (5) ate the idenlification mappings in (11) and (12). P r o o f. We compare the equations (10) and in Proposition 6, and see the difference at a3 and icr3.…”

Section: (G74mentioning

confidence: 96%

Hurwitz duality theorems for Fueter and Firac equations

Ławrynowicz

Сузукі

1997

AACA

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“…Let us analyze the situation more closely. In [26], we can find such dualities, expressed as type-changing transformations, relating manifolds M with the following pseudometrics: By analogy with Penrose twistors [27], we arrive at the structure of Hurwitz twistors (H for short) [28] determined by a system of 5 4 = 5 algebraic equations and at their antiobjects (aH) corresponding to the fivedimensional Kałuża-Klein theory [29,30]. 2 9 and dx 2 1 + .…”

Section: (15)mentioning

confidence: 99%

“…The algebraic content of this stuff is strictly related to the idea of one of us and Rembieliński [29,30] to consider two vector spaces, the so-called Hurwitz pair (S, V ), restricted by the Hurwitz-type condition:…”

Section: Geometrical Model Of the Surface Melting Effectmentioning

confidence: 99%

Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction

2008

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The idea of fractal modeling of crystals dates back to Bethe [1], who observed its convenience when coming to first, second, and third nearest neighbors of an atom. Taking into account that it is a neighbor of two or more other atoms, even in the case of one layer with a lattice formed by squares, one naturally comes to the notion of cluster [2,3]. It is then natural to cut the plane of the lattice correspondingly to the cluster involved and construct a Riemann surface or a Bethe lattice-a fractal set of the branch type [4,5]. The construction is parallel to that related to the holomorphic function f (z) = exp z 2 in C (Fig. 1). The example already shows the importance of the corresponding group Γ of cover symmetry transformations (Decktransformationengruppe), inoculation (of the branch corresponding to no. 1 on the branch corresponding to no. 4), and gradation related to the points , •, , and •.The next important step was done by Kikuchi [6], who-within his theory of cooperative phenomenadeveloped a method of approximation for order-disorder phenomena.In this context, Sukiennicki, Wojtczak, Zasada, and Castillo Alvarado [7] investigated an infinite thin film of an AB 3 alloy. As examples we may take Ni 3 Fe or Cu 3 Au. They assumed the (111) orientation of the alloy. Let z(j) denote the concentration of A-atoms in the layer j = 0, 1, . . . , where j = 0 corresponds to the surface. Then the concentration of B-atoms in that layer is 1 − z(j). If U is the energy of interaction of the system, T is the absolute temperature, and g is the number of possible configurations, then the entropy S and the free energy F of the system are given byrespectively, where k is the Boltzmann constant, and the conditions for thermodynamic equilibrium at a given temperature T are ∂ ∂τ F τ =τ (j) = 0, ∂ ∂z F z=z(j) = 0 for each j, λ = const. Thus, 1 T ∂ ∂τ U τ =τ (j) = ∂ ∂z S τ =τ (j).

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“…Still, following Kaluza and Klein [8], gravity and electromagnetism can be unified by making use of five dimensions which are the four-dimensional spacetime M 0 and the one-dimensional space which, compactified, as S l5 gives rise to a 1/(1) gauge symmetry [9]. Following [10], Sect. 6, consider the Clifford matrices y ß , p= 1, 2, 3, 0, corresponding to the purely imaginary Majorana representation of the Clifford algebra C 13 ' 1 '.…”

Section: The Dirac Field and The Electromagnetic Fieldmentioning

confidence: 99%

“…As remarked in [10], S fully determines the transformations shown by arrows in the list of four possible mathematical descriptions of the field in question:…”

Section: The Dirac Field and The Electromagnetic Fieldmentioning

confidence: 99%

Fields Connected with Particles in Terms of Complex-Riemannian Geometry

Ławrynowicz

Kędzia

Wojtczak

1990

Zeitschrift Für Naturforschung A

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A complex analytical method of solving the generalised Dirac-Maxwell system has recently been proposed by two of us for a certain class of complex Riemannian metrics. The Dirac equation without the field potential in such a metric appeared to be equivalent to the Dirac-Maxwell system including the field potentials produced by the currents of a particle in question. The method proposed is connected with applying the Fourier transform with respect to the electric charge treated as a variable, with the consideration of the mass as an eigenvalue, and with solving suitable convolution equations. In the present research an explicit calculation based on linearization of the spinor connections is given. The conditions for the motion are interpreted as a starting point to seek selection rules for curved space-times corresponding to actually existing particles. Then the same method is applied to solids. Namely, by a suitable transformation of the configuration space in terms of elements of the interaction matrix corresponding to the Coulomb, exchange, and dipole integrals, the interaction term in the hamiltonian becomes zero, thus leading to experimentally verificable formulae for the autocorrelation time

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Pseudo-Euclidean Hurwitz pairs and the Kaluza-Klein theories

Cited by 27 publications

References 11 publications

Hurwitz duality theorems for Fueter and Firac equations

Hurwitz duality theorems for Fueter and Firac equations

Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction

Fields Connected with Particles in Terms of Complex-Riemannian Geometry

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